The displacement of a plane progressive wave | vedclass

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Question

(A) The general equation for a wave travelling in the positive $x$-direction is $y = A \sin(\omega t - kx + \phi)$.At $t=0$, the equation is $y = 3 \s

Vedclass · Accepted Answer

$y=3 \sin 2 \pi \left(-\frac{x-16}{3}\right)$

Vedclass · Answer

(A) The general equation for a wave travelling in the positive $x$-direction is $y = A \sin(\omega t - kx + \phi)$.At $t=0$, the equation is $y = 3 \sin(-2\pi x / 3)$, which implies $k = 2\pi / 3$.Given the wave velocity $v = 4 \ m/s$, we use the relation $v = \omega / k$.Thus, $\omega = v \cdot k = 4 	imes (2\pi / 3) = 8\pi / 3$.The general wave equation is $y = 3 \sin \left(\frac{8\pi}{3}t - \frac{2\pi}{3}xight)$.At $t=4 \ s$, substituting the value of $t$:$y = 3 \sin \left(\frac{8\pi}{3}(4) - \frac{2\pi}{3}xight)$$y = 3 \sin \left(\frac{32\pi}{3} - \frac{2\pi}{3}xight)$Factoring out $2\pi$:$y = 3 \sin 2\pi \left(\frac{16}{3} - \frac{x}{3}ight) = 3 \sin 2\pi \left(-\frac{x-16}{3}ight)$.

The displacement of a plane progressive wave in a medium, travelling towards the positive $x$-axis with a velocity of $4 \ m/s$, at $t=0$ is given by $y=3 \sin 2 \pi (-x/3)$. The expression for the displacement at a later time $t=4 \ s$ will be:

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